Instructor: Joshua Reyes
Time: M Th 5:00PM  6:30PM
Start Date: 4 June 2007
Location: Taffee Tanimoto Conference Room (S3180)
Email: jreyes@removeme.post.harvard.edu
Recommended Reading:
The graphic for the poster came from a comic by Kevin de Leplante called How to Build Your Own Wormhole (Suitable for Interstellar Travel). In fact, the comic inspired the layout and title for the poster, too. Almost nothing I do is ever my own.
Tenative syllabus [PDF]. Poster [PDF].
If there is something that excites you about differential geometry or general relativity, I'll try to work it into the seminar (to the best of my abilities). Please do not hesitate to send me an email with any questions, comments, concerns, anecdotes, or other short stories.
You should not feel like you have to finish all the reading assignments. Each of the sources approaches the material in wildly different ways and levels of rigor. Try to find a source that suits you. And let me know what you think of the texts. I'm not following any one source all that closely (though if I had to pick one, I guess it'd be O'Neill), so look around.
Lecture  Date  Exercises  Reading 
1  4 June  Draft [PDF] [DVI]  
7 June  No Class.  
2  11 June  Review the derivative from multivariable calculus. In particular focus on the chain rule and product rule. To get you going, here's something to think about. Consider the map T on Mat(n, n), the space of n×n matrices with real coefficients, taking a matrix A to its square A^{2}. What is the derivative D(T(A)) when n = 1? What is it when n > 1?  Hubbard and Hubbard chapter 1 
3  14 June  Show that any open set of a manifold is, itself, a manifold by constructing an altas for it explicitly.  
4  18 June  Tangent vectors, vector fields, differentials. Show that differential dx^{i}(V) does, indeed, select the ith coordinate from the vector field V. Show that the dx^{i} form a dual basis to the partial derivatives ∂_{i} according to the definitions. That is, dx^{i}(∂_{j}) is 1 if i=j and 0 otherwise.  
5  21 June  Look up the statement of the inverse function theorem, peak at a proof if you feel like it. Read about submanifolds, especially submersions. For those who want to put everything in R^{n}, check out Whitney's embedding theorem.  Guillemin and Pollack, Chapter 1.3–1.4 
6  25 June  Review multilinear algebra: in particular, know what a module is.  O'Neill, first half of Chapter 2; Boothby, Chapter 5; Wald Chapter 2.2–2.4; Chapter 5 in d'Inverno could be helpful. 
7  28 June  Derive the transformation law for oneforms using the coordinate expression for the differential of a smooth map between manifolds and the definition of a dual basis. Use the definition of the tensor product to derive the transformation law for a general tensor of type (r, s).  Boothby, Chapter 7.1–7.4; O'Neill, second half of Chapter 3 (scalar products), an assortment from Chapter 4; Wald 3.1–3.3; d'Inverno, some of Chapter 6 
8  2 July  Show that the set of isometries on a semiRiemannian manifold (M, g) forms a group; i.e., the identity map is an isometry, the composition of two isometries is an isometry, and the inverse of an isometry is an isometry. (Recall: a smooth map φ: M→ M is an isometry if it is a diffeomorphism such that φ^{*}(g) = g.)  
5 July  No class.  
9  9 July  For a null vector n and another vector w that is not orthogonal to n, show that the operator h = g + w⊗n/⟨w, n⟩ is a projection operator, but is not symmetric.
I.e., show that i) h(n) = 0,
ii) h(v) = v if and only if ⟨v, n⟩ = 0,
iii) that h^{2} = h(h) = h, and
iv) tr h = dim M1. (Hint: Use local coordinates: h^{α}_{β} = δ^{α}_{&beta}+w^{α} n_{β}/(∑ w^{μ}n_{μ}) Justify why you can do the calculations locally.) 

10  12 July  Define the curl of a vector field V by its action on two other vector fields X and Y, namely, curl V(X, Y) = ⟨D_{X} V, Y⟩ − ⟨D_{Y} V, X⟩. Show that curl V is an antisymmetric (0, 2)tensor. (Such a creature is called a 2form.) Find its components relative to a local coordinate chart. (Hint: Express V in local coordinates and let X = ∂_{i} and Y = ∂_{j}.) 

11  16 July  Let D be a tensor derivation. Let its action on the basis vectors in a coordinate chart be D(∂_{i}) = ∑ F^{j}_{i} ∂_{j}. Show that for the oneform dual basis that D(dx^{j}) = −∑ F^{j}_{i} dx^{i}. (Notice that the indices i and j have been swapped.) (Hint: Using the axioms of a tensor derivation, apply the derivation to the expression θ(X) = C(θ⊗X), where θ is a oneform, X is a vector field, and C is a contraction map, to find the action of a derivation on a oneform in terms of its action on vector fields and functions.)  
12  19 July  Let D be the LeviCivita covariant derivative on a semiRiemannian manifold (M, g). Show that the metric tensor g is parallel with respect to D; i.e., D_{V} g = 0 for all vector fields V. (Note: this is equivalent to axiom of metric compatability presented in lecture.)  
13  23 July  1. Prove that if a geodesic is timelike at a given point p, then it is timelike everywhere along its length. Prove similar
results for null and spacelike geodesics. 2. For the 2dimensional metric ds^{2} = (dx^{2} − dt^{2})/t^{2}, find all the Christoffel symbols Γ_{ijk} [only four of them are nonzero], and find all the timelike geodesic curves. 

14  26 July  Calculate the Riemann curvature tensor for a semiRiemannian manifold (M, g) for the cases dim M = 1, 2, or 3.  
15  30 July  1. Let G = Ric − S g/2 be the Einstein curvature tensor, Ric be the Ricci curvature tensor, S be scalar curvature, and g is the metric tensor of a 4dimensional semiRiemannian manifold. Prove that Ric = G − tr(G) g/2.
2. A Riemannian manifold (M, g) is called Einstein if the Ricci curvature of M is a scalar multiple of the metric, Ric = cg, for some constant c. Show that for dim M ≥ 3, Ric = λg for some function λ implies the manifold is indeed Einstein. (Hint: Use divergence to get one constraint on λ. Apply contraction to get another.) 

16  2 August  1. Show that [X, Y] = D_{X} Y − D_{Y} X completes our proof that the Hessian of f, H ^{f}(X, Y) = D(Df)(X, Y) = XYf − D_{X} Yf, is symmetric. (Recall: We left it at H ^{f} = YXf − D_{Y} Xf.) Demonstrate C^{∞}linearity in Y to prove that it is a (0, 2)tensor. (Why only in Y?) Go on to show that H ^{f}(X, Y) = ⟨D_{X} grad f, Y⟩. (Hint: Remember that grad f is the vector that is metrically equivalent to df. Combine that fact and compatibility of the covariant derivative with the metric to prove the result.)
2. Define the curl of a vector field V as in Lecture 10 above. Show that curl (grad f) = 0. 

17  6 August  Prove (from the definition) that the wedge product is distributive. Namely, show that (cα + β)∧&gamma = c(α∧γ) + β∧&gamma, for any constant c and differential forms α, β, and γ  
18  9 August 
1. Prove that vectors v_{1},…,v_{k} are linearly independent if and only if v_{1}∧⋅⋅⋅∧v_{k} ≠ 0. (Hint: You may find it useful to extend the set of vectors to a basis.) 2. Let v ∈ V be a nonzero vector, and let w ∈ ∧^{k} V^{*}. Prove that v∧w = 0 if and only if there exists u ∈ ∧^{k − 1} V^{*} such that w = v∧u. 